©<& 


UC.NRLF 


^^aaraa^ak^a^i^y-?--^-^ 


m  MEMOMAM 
Edward  Bright 


ACADEMIC 


TRIGONOMETRY. 


PLANE  AND  SPHERICAL. 


BY 

T.  M.  BLAKSLEE,  Ph.D.  (Yale), 

Professor  of  Mathematics  in  the  University  of  Des  Moines. 


3j*JC 


BOSTON: 

PUBLISHED  BY   GINN  &  COMPANY. 

1888. 


J?9>,. 


Entered,  according  to  Act  of  Congress,  in  the  year  1»87,  by 

T.  M.  BLAKSLEE, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


Typography  by  J.  S.  Cushino  &  Co.,  Boston. 
Pkbsswobk  by  GiNN  &  Co.,  Boston. 


PREFACE. 


r  I  ^HE  purpose  of  this  arrangement  is  to  aid  the  memory 
by  noting  analogies. 

+  and  a^  have  as  spherical  analogies  x  and  cos  a.     Page 
12  has  page  13,  and  each  "Law"  has  its  analogy. 

In    Spherical    Trigonometry    we    note    the    determining 
groiqjs:   side,  +,  co.  function,  and  Z,   — ,  function. 

It  is  hoped  that  the  Introduction  will  fix  the  character- 
istics of  Trigonometry. 

This    should    be    accompanied    by   practical   work,   and 
occupy  at  least  a  week. 

T.  M.   BLAKSLEE. 
Des  Moines,  Ia. 


Note.  It  is  convenient  for  examinations  to  have  tables  separate 
from  formulas. 

The  explanation  of  the  use  of  tables  should  be  with  them. 

Two  pages  of  model  solutions  and  answers  may  be  added.  (Opin- 
ions asked  on  this  point.) 


797961 


Digitized  by  tine  Internet  Archive 

in  2008  witii  funding  from 

IVIicrosoft  Corporation 


littp://www.arcliive.org/details/academictrigonomOOblakricli 


INTKODUCTION. 


Def.  Trigonometry  is,  etymologically,  the  Science  of  Meas- 
uring Triangles.  Besides  this,  it  now  inchides  the  Science  of 
Angular  Functions. 

We  first  inquire,  What  is  a  function?  then.  What  are  the 
angular  functions? 

A  function  of  a  variable  is  a  second  variable  so  related  to 
the  first  that  any  change  in  the  variable  produces  a  change  in 
the  function. 

III.   Oil  in  lamp  and  time  it  has  burnt. 

Def.  The  functions  of  the  angle  between  two  straight 
lines  are  the  six  ratios  of  the  sides  of  the  right  triangle  formed 
b}"  these  lines  and  a  perpendicular  upon  one  of  them  from  a 
point  in  the  other. 

Notation,   /i,  hypotenuse  ;  o,  opposite  ;  a,  adjacent. 

h^ 


The  ratios  are,  by  definition, 


sine        =  sin  =  -• 
h 

.-.  0  =  /isin. 

sin 

cosine    =  cos  =    • 
h 

.-.  a  =  /icos. 

cos 

tano;ent  =  tan  =  -• 
a 

,'.  0  =  rttan, 

tan 

And  their  reciprocals. 

sin       0 
cos      h 

a      0      , 
-  =  -==  tan. 
h      a 

8 


ACADEMIC   TRIGONOMETRY. 


By  similar  triangles,  the  functions  are  constants  for  a  constant  angle, 
but  variables  for  a  variable  angle. 

The  base  line  is  the  initial  line ;  the  hypotenuse  line,  the  terminus  of 
the  angle. 

Linear  Representation.  (1)  If  h=l,  o  =  sin,  a  =  cos. 
(2)  If  a=  1,  o  =  tan. 

The  transverse  line  is  TT'  through  vertex  and  perpendicu- 
lar to  initial  line  //'. 

sin  =  transverse  projection  of  directed  unit.    (Unit  7i.) 

cos  =  initial  projection  of  directed  unit.* 

tan  =  transverse  projection  of  h  if  initial  projection  be  unity.f 
Since  antecedent  =  consequent  x  ratio,  also 
For  sine  and  cosine,  consequent  =  /i,  ratio  =  function. 

Rule  I.  To  obtain  either  side  from  /<,  multiply  by  ratio, 
sine  for  o,  cosine  for  a. 

Rule  II.  To  obtain  the  sine  from  cosine,  multiply  by 
tangent. 


T'-\ 

h- 

T- 

II 

TA 

r    t\ 

h 

I 

I 

a 

a 

> 

0 

\^ 

h^ 

1'.- 

— 

V"  ;. 

^ 

a 

a 

-h 

r  "'ijT 

T 

T 

T 

Quadrants .    IV  and   TT'  divide  the  angular  space  about 
the  vertex  into  four  quadrants,  numbered  as  in  the  figure. 
An  angle  is  in  the  quadrant  in  which  it. terminates. 

*  The  angle  being  the  direction  of  its  terminus,  we  may  speak  of  the 
ratios  as  direction  ratios. 

Since  for  the  other  acute  angle  of  ratio  triangle, 

sin  =  -,  cos  =  -,   and  tan  =  -  =  cotangent  =  cot. 
h  h  0 

.'.  "co"  means  oj"  complement. 
t  If  a  circle  be  described  with  the  unit  base  a  as  radius,  o  is  a  tangent. 


INTRODUCTION. 


The  Terminal  Values  of  the  functions  are  as  follows  : 


I. 

II. 

m. 

IV. 

z 

0°. 

90^ 

180^ 

270°. 

sin 

+ 

+ 

- 

- 

sin 

0 

+  1 

0 

-1 

cos 

+ 

- 

- 

+ 

cos 

+  1 

0 

-1 

0 

tan 

+ 

- 

+ 

- 

tan 

0 

GO 

0 

GO 

The  algebraic  signs  being  determined  thus :    to  right  and 
up,  +  ;  to  left  and  down,  — . 


PRACTICAL  DEVELOPMENT. 

Wishing  to  calculate  the  distance  IB  to  an  object  .B,  start- 
ing from  /,  I  laid  off  lA  ±  IB. 

At  a  distance  AM=:::  1  from 
A  I  erected  MN  _L  AI,  deter- 
mining N  by  looking  from  A 
to  B. 

I  also  measured  AP,  and 
drew  PL  ±  AB. 

The  last  is  not  needed  in  measuring  the  distance  ;  in  fact, 
AM  might  have  been  any  distance,  when  IB  could  have  been 
MNxAI 


unit 


L  M 


found,  as  IB  = 


AM 


The  advantage  of  a  table  of  tangents  is,  that  we  never  have 
need  to  construct  the  small  triangle. 

If  lA  =  1000  feet,  and  we  have  the  tangent  from  a  table, 
we  have  simply  to  move  the  decimal  point  three  places,  and 
we  have  IB  at  once. 

Two-Place  Table.  Take  10  inches  as  an  hypotenuse,  and, 
by  aid  of  a  protractor  (or  by  consti'ucting  an  angle  of  30°, 
geometrically,  and  then  trisecting  it  by  folding),  construct  the 


10 


ACADEMIC    TRIGONOMETRY. 


values  of  sine  and  cosine  f  .-.  tan  =  ^  )  for  every  10°.    Here 

\  cosy 

10  inches  =  unit.     .-.  0.1  inch  =  0.01  unit. 


Evidently  (arithmetically)   function  (180°  -  ^)  = /(.4). 
The  ratio  triangles  being  equal,  having  h  and  A  equal. 


EXAMPLES. 


z° 

10 

20 

30 

40 

50 

60 

70 

80 

sin 

17 

34 

50 

64 

77 

87 

94 

98 

cos 

98 

94 

87 

77 

64 

50 

34 

17 

tan 

18 

30 

58 

84 

1.19 

1.73 

2.75 

5.67 

1.  Give  functions,  if  o,  a,  A,  are  (1)  6,  8,  10;    (2)  10, 

24,  26;   (3)  4,  7,  5,  8.5. 

2.  Solve  the  following  :  Z,o,aJi  being  (1)  20°,  ?,  ?,  100  ; 
(2)   ?,  4,  ?,  5;   (3)  57',  4000,  ?,  ?;   (4)  8.8",  4000,  ?,  ?. 

Note.  If  the  greatest  angular  distance  of  Venus  from  the  sun  be 
45°,  what  is  its  distance  from  that  body  as  compared  to  that  of  the 
earth  ? 

3^  Canthe  sines  of  0°,  30°,  45°,  60°,  and  90°  be  written, 

iVo,  iVT,  iV2,  |V3.  |V4? 


INTRODUCTION.  11 


4.    If  A,  B,  and  C  be  the  angles,  and    a,  5,  and  c  the 
opposite  sides  of  a  triangle,  p  the  perpendicular  from  C  to  c, 
show  that  a  sin  jB  =  ^  =  ?>  sin  ^.      .- .  a  :  b  =  sin  A  :  sin  B. 
(In  words.) 
Do  field  work,  using  ratios  to  two  places, 
sinl5  =  i(0.17  +  0.34), 
sin  18  =  sin  10  +  0.8  x  0.17  =  0.31. 

Note.  If  the  greatest  and  least  values  of  the  maximum  elongation 
of  Mercury  be  15°  and  30°,  what  are  its  greatest  and  least  distances 
from  the  sun  ? 

Logarithmic  Solutions.  Though  strictly  Algebra,  we  give 
the  logarithmic  solutions  thus  far : 

log  of  sin  =  log  0  —  log  7i.    .  • .  log  o  =  log  h  +  log  of  sin. 
log  of  cos  =  log  a  —  log  h.    .  • .  log  a  =  log  h  +  log  of  cos. 
log  of  tan  =  log  0  —  log  a.    .  • .  log  o  =  log  a  +  log  of  tan. 
log  sin       =  log  of  sin  +  10,      a:b  =  sin  A  :  sin  B,  gives 
log  sin  A  =  log  a  -f  colog  b  +  log  sin  B. 


12 


ACADEMIC   TRIGONOMETRY. 


PLANE. 


THE   DEFINING  EQUATIONS. 


(1)  By  T,,  smA 

(2)  By  Tg,  cos^ 

(3)  By  7^2,  tan^ 


a  =  h  sin  A, 
b  =  h  cos  yl, 
a  =  b  tan ^4, 


h 


sin^ 
b  _ 

cos^ 
a 


a 

b'  '  tan^ 

(1)  is  the  definition  of  the  sine  ratio. 

(2)  is  the  definition  of  the  cosine  ratio. 

(3)  is  the  definition  of  tlie  tangent  ratio. 
By  T;,  sin2  +  cos2=l. 

.♦.  a^  =  Ji^ sin^,  and  b^  =  Ir  cos^. 
...   (4)i/i2  =  a2-f-6';   (4)2  sin2-hcos2=l=cot^cotB. 

(4)i  is  the  Pythagorean  formnla. 

(5)   By  7^4,  sin  ^  =  cos  5,  cos  ^  =  sin  5. 

(5)   is  the  complementary  formnla. 
Note.   For  construction  of  figure,  p.  13,  see  p.  15. 


THE  FUNDAMENTAL  ANALOGIES. 


13 


C.B. 


SPHERICAL. 


tan.  b 


THE   FUNDAMENTAL  ANALOGIES. 


/ 1  \  T^    m      •     A      Sin  a      .  •    T    •      4      '    1       sin  a 

(1)  By  Ti,  sin  A  = ,  sin  a  =  sin  h  sin  A,  sin  h  = —- 

sin  h  sin  ^ 


(2)  By  T3,  cos^  =  ^^,  tan& 
^  ^     *^  tan /a 


tan/i  cos^,  tan/i  = 


(3)  By  T,,  HnA 


tan  a 


tan  a  =  sin  6  tan  A,  sin  6  = 


tan?> 
cos^ 
tana 
tan  J. 


sin  6 

(1)  is  the  sine  analogy,  or  s'ui  Ay. 

(2)  is  the  cosine  analogy,  or  cos  Ay. 

(3)  is  the  tangent  analogy,  or  tan  Ay. 

By  T'4,  VAiIIi,  cos  h  =  cos  a  cos  6,  by  (3) . 

(4)  cos  7i  —  cos  a  cos  6  =  cot  A  cot  ^.  -.iiJi 
The  first  is  the  Pythagorean  analogy. 

Dividing  sin^  (1)  by  cosJ5  (2),  nsing  (4). 

(5)  sin  A  —  cos  B :  cos  6,  sin  B  =  cos  A  :  cos  a. 

(5)  is  the  complementary  analog}'. 

Note.  If  a  sphere  of  unit  radius  be  described  about  Fas  a  centre, 
the  faces  will  cut  out  a  right  spherical  triangle  having  tlie  sides  a,  h, 
and  //. 


fi, 


14 


ACADEMIC  TKIGONOMETKY. 


NEGATIVE  ANGLES. 

Note.  A  point  moving  to  the  right  generates  a  +  distance,  but 
moving  back  to  the  left  tends  to  destroy  this,  and  passing  the  origin 
generates  a  —  distance. 

A  straight  line  revolving  in  the  order  of  the  quadrants  I,  II,  III,  IV", 
generates  a  +  angle,  but  revolving  back  tends  to  destroy  this,  and  pass- 
ing the  initial  line  generates  a  —  angle. 

Rule.    Changing  the  sign  of  an  angle  changes  the  sign  of 
its  sine,  but  not  of  its  cosine. 
.'.  changes  that  of  its  tangent. 


+z 

I. 

II. 

III. 

IV. 

-z 

-I. 

-II. 

-III. 

-IV. 

sin 

-f 

+ 

- 

- 

sin 

- 

- 

+ 

+ 

cos 

+ 

- 

- 

+ 

cos 

+ 

- 

- 

+ 

tan 

+ 

- 

+ 

- 

tan 

- 

+ 

- 

+ 

Since  the  terminus  is  changed  across  the  initial  line  II', 
but  not  across  the  transverse  line  TT'. 

Thatis, /'=IV,  7/'=  III,  III'=U,  IV'=J. 


FUNCTIONS   OF  nr±A. 

Rule.  If  an  acute  angle  be  added  to  or  subtracted  from 
an  even  number  of  quadrants,  the  functions  of  the  resulting 
angle  are  equal  arithmetically  in  value  to  the  like-named 
functions  of  the  acute  angle  ;  but  if  an  acute  angle  be  added 
to  or  subtracted  from  an  odd  number  of  quadrants,  the  func- 
tions of  the  resulting  angle  are  arithmetically  equal  in  value 
to  the  co-named  functions  of  the  acute  angle. 

By  the  equality  of  the  eight  possible  ratio  triangles,  and 
the  fact  that  for  an  even  number  of  quadrants  a  and  o  are 


CONSTRUCTION. 


15 


the  same  as  for  A,  but  for  an  odd  number  they  are  inter- 
changed. 


Note.  By  revolving  1^  and  1^^  through  any  number  of  right  angles, 
one  rotation  changes  sine  to  cosine,  two  restores,  and  so  on. 


CONSTRUCTION. 

(1)  Lay  off  from  the  vertex  Fof  a  right  trihedral  a  unit 
on  each  edge  (F// being  edge  of  rt.  Z). 

(2)  Through  the  extremity  of  one  of  the  acute  edges,  as  ^i, 
pass  a  plane  ±  to  the  other  acute  edge  VA^  thus  : 

Draw  BH±  VH,  then  HA  ±  VA,  lastly  join  AB. 
(By  Geom.)  BAH  is  the  plane  measure  of  dihedral  having 
edge  VA. 

(3)  Through  the  otlier  extremities  H^  and  ^3,  pass  planes 
II  \joA,BJI,,  .-.  ±  to  VA. 

(4)  By  p.  12,  the  parts  of  the  nine  right  triangles  are  as 
given. 

Note.  Napier's  Circular  Parts  are :  Tlie  two  sides  about  the  right 
angle,  the  complements  of  the  opposite  angles,  and  the  complement  of 
the  hypotenuse. 

His  rules  are : 

Rule  I.  The  sine  of  the  middle  part  is  equal  to  tlie  product  of  the 
tangents  of  the  ar^jacent  parts. 

Rule  II.  The  sine  of  the  middle  part  is  equal  to  the  product  of  the 
cosines  of  the  opposite  parts. 


16 


ACADEMIC   TRIGONOMETP.Y. 


By  I  sin  a  =  sin  A  sin  h  =  tan  b  cot  B 


(1)1  sin  6  =  sin  ^  sin  A: 
cos^=  sin  B  cosa 


tan  a  cot  A 
tan  b cot  A 


(3) 


(5)  ""'' "^-  ""•  ^  ""'^ ^  -  ^""  "  """ "   (2) 
I  cos  J5=  sin  ^  cos  6  =  tan  a  cot  A  | 

(4)  I  cos  A  =  cos  a  cos  6  =  cot  ^  cot  i?  1(4) 

I.  By  (Comp,  Ay.)  An  oblique  angle  and  its  opposite  side  are  in 
the  same  quadrant. 

II.  By  (P.  Ay.)  h  <  90°  when  a  and  b  are  in  the  same  quadrant. 

h  >  90°  when  a  and  b  are  in  different  quadrants. 


EXAMPLES  INVOLVING   RIGHT  TRIANGLES. 

Solve  in  order  of  formulas.     Each  triangle  furnishes  nine 
examples. 


5,C,? 

s,1,t 

1,c,t 

s,c,  t 

?,?,  ? 

h 

a 

b 

A 

B 

1. 

10.40 

5.43 

8.87 

31°  30' 

58°  30' 

s  =  sin  Def . 

2. 

37.57 

27.81 

25.26 

47°  45' 

42°  15' 

c  =  cos  Def. 

3. 

7855. 

6967.8 

3627.4 

62°  30' 

27°  30' 

t  =  tan  Def. 

4. 

50 

30 

40 

36^  52'  10" 

53°    7' 30" 

P  =  Pyth.  Theo. 

5. 

13 

5 

12 

22°  37'  10" 

67°  22'  50" 

Comp.=co-relation. 

6.  The  distance  of  the  moon  being  7i,  and  earth's  radius  a 
A  =  57'  2"  ;  find  h.     For  the  sun,  A  =  8.8". 

7.  What  is  the  length  of  the  horizontal  shadow  of  the 
Washington  Monument,  when  the  altitude  of  the  sun  is  50^? 

8.  What  is  the  radius  of  the  circle  of  latitude  on  which 
you  live  ? 

9.  (a)  The  angle  of  elevation  of  the  top  of  a  spire  500 
feet  distant  is  measured  and  found  to  be  13°.  What  is  its 
height  [above  the  instrument]  ? 

(h)  The  elevation  of  base  of  spire  is  9°.  What  is  its 
height  ? 


EXAMPLES. 


17 


10.    The  elevation  of  the  top  of  a  spire  is  45°,  and  at  a 


point  100  feet  farther  away,  36°  52'. 


What  is  the  height? 


y 


t-V 


Spherical  Right  Triangles.  First  find  a  from  A  and  /i,  then 
A  from  a  and  li.  That  is,  solve  in  the  order  that  the  formu- 
las are  given  (p.  13). 


a 

b 

h 

A 

B 

1. 

20°  4' 22" 



59°  33'  43" 

23°  27'  29" 

S.Ay. 

2. 

12°  56'  43" 

34°  14'  45" 

23° 27' 31" 

3. 

57°  21' 33" 

59°  33'  43" 

23° 27' 29" 

C.Ay. 

4. 

43°  20' 58" 

44°  9' 38" 

13°  34'  30" 

5. 

45°  7' 49" 

85°  45'  2" 



45° 12' 33" 

T.  Ay. 

6. 

38°  1'40" 

139°  24' 

50°  14'  8" 

7. 

22°  7' 15" 

69°  30'  13" 

71°  4' 20" 

P.  Ay. 

8. 

2°  59' 

22°  21'  51" 

22°  33'  12" 



9. 

43°  35' 

46°  59' 

58°  50' 

Comp.  Ay. 

10. 

22°  15'  7" 

27°  28'  38" 

73° 27' 11" 



The  right  ascension  i2,  declination  cZ,  and  longitude  Z,  of 
the  sun  form  a  right  triangle  of  which  these  are  6,  a,  and  h ; 
A  being  the  obliquity  of  the  ecliptic. 

11.  i  =  214°  14' 45";  find  i2  and  d. 

12.  72=  18  hrs.  44  min.  50  sec.  ;  find  L  and  d. 

13.  i2  =  4hrs.  38  min.  0.88  sec,  d  =  22°  7' 13.7";  find 
L  and  A. 

REMARKS    AND    QUESTIONS. 

Many  points  rest  directly  upon  page  12.  Thus  pages  13, 
14,  18,  19,  and  23  in  great  part. 

As  the  last  of  24,  and  most  of  25,  require  19  and  20,  the 
given  order  has  been  followed. 

It  is  very  important  for  the  student  to  observe  as  to  what 
rests  directly  on  pages  12  and  13. 


18  ACADEMIC  TRIGONOMETRY. 

Give  the  values  of  the  functions  of:  60°,  120°,  150°,  225°, 
-30°,  -60°,  -1210°,  350°,  440°,  1000°,  1234°. 

How  are  the  functions  of  115°  related  to  those  of  205°? 

Reduce  {x^  +  y^)cos  720°-  2xy  sin  540°. 

When  is  it  sufficient  to  consider  angles  in  (1)  I,  (2)  I  and 
II,  (3)1, -IV,  (4)V, -VI,  ? 

VALUES  OF  ONE  FUNCTION  IN  TERMS  OF  ANOTHER. 

By  P.  Theo.,  V  =  sin^  +  cos^  =  1.  (1) 

.-.  sin=  Vl  —  cos^,  cos  =  Vl  —  sin''^. 

Note.   Prove  this  without  P.  Theo.,  and  thus  prove  the  theorem. 

By  P.  Theo.,  f-^\  =  rec.  cos^  =  tan^  +  1. 

\cosy 

1  .         ,  tan 

. • .  cos  =  — =zi=:^ ,  sin  =  tan  cos  =  —  -• 

Vl  +  tan^  a/1  +  tan^ 

Give  the  value  of  each  function  in  terras  of  the  others. 

1 .  sin  30°  =  ^  ;  find  the  other  functions. 

2.  cos  45°  =  I V2  ;  find  the  other  functions. 

3.  tan ^1=  1,  2,  3,  ...  ;  find  otlier  functions. 

Hint.    If  tan  =  3,  rec.  cos^  =  10.     .-.  cos  = ,     sin  =  — ^• 

VTO  VlO 

4.  Given  one  function  of  90°  ;  find  the  others. 

5.  (1)  2  sin  =  cos  (2)  tan  =  2  sin. 
(3)      sin  =  COS.  (4)  sin  =     tan. 

Hint  on  (2).  —  =  2  sin.    .'.  cos  =  I.   Z=  60°. 
cos 

Hint  ON  (1).  sin  =  r,   2x=  Vl  —  x-.     .-.  a:  =  sin  =  ^V5. 

6.  cosnan2  +  sin2cot2=l. 

7.  In  rt.  sph.  A,  sin^/i  =  sin^o  +  sin^a  [cos^o]  (2d P.  Ay.). 

=  sin-a  H-  sin^o  [cos^a]. 
Its  limit,  /r  =  a"  +  o^ 


FUNCTIONS   OF    SUM. 


19 


(a)    FUNCTIONS   OF    SUM. 


■-^^A 

COS.  A  COS.  B.—sin.A  sin..B. 

Directly  from  figure  and  page  12, 

sin  (^  + -B)  =  sin  ^  cos  JB  +  cos  ^  sin  ^.  (1) 

cos  (^  4- B)  =  cos  J.  cos  5  —  sin  ^  sin  ^.  (2) 

Dividing  sine  by  cosine,  then  both  terms  of  fractions  by 
cos  A  cos  B, 


tan(J[  +  iB)=*^^^^i±i^^. 
l-tanu4tanJ5 

(6)  Double  Angle.     IfA  =  B. 
sin  2  J.  =  2  sin  A  cos  A. 
cos  2A  =  cos^A  —  s'm^A. 
2tan^ 


tan  2  A 


l-tan^^ 


(c)  Sum  and  Difference  of  Sines.    If  A-\-B 
A-B=D. 


(3) 

(1) 
(2) 

(3) 

S,   and 


A  =  ^  +  ^,       B  =  ^-^. 
2^2                  2        2 

sin  ^  =  sin  —  cos  --  -f-  cos  —  sin  — 

sin  5  =  sin  f  cos  :5  _  cos  -  sin  - 
2         2             2         2 

sin^  +  sinJB=2sin-cos- 
2         2 

(1) 

sin  ^  —  sin  -B  =  2  cos—  sin  — 

9              9 

(2) 

20  ACADEMIC   TRIGONOMETRY. 


(a)  FUNCTIONS  OF  DIFFERENCE. 

sin  (A-B)  =  BinlA  +  ( -5)] 

=  sin^4cos(— 5)  +  cos^siii(— i^). 
By  rule  for  negative  angles,  page  14, 

sin  (A  —  B)  =  sin  ^  cos  JB  —  cos^  sin  jB.  (1) 

cos(^  — -B)  =  cos^cos  JB-f  sin^sin^.  (2) 

tan(^-^)=^^^^-;^^-^.  (3) 

^  ^      1  +  tan^tanjB  ^ 

(6)  Half  Angle.   By  double  angle  formulas, 

COS"*  — h  sin''—  =  1 . 
2  2 

cos^ sin^— =  cos^. 

2  2 

.*.  having  the  sum  and  the  difference  of  sin^  and  cos^. 


.    A         1  — cos^l  x^x 


A 

cos  — 

2 


=aI^^^-  (2) 


''CO   +  sm 


tan^  =  Ji^^^.  (3) 

2       \l  +  cos^  ^  ^ 

(c)  Sum  and  Difference  of  Cosines. 

,  S      D       .   S  .  D 

cos  A  =  cos  —  cos sin  -  sin — 

2         2  2        2 

„  S       D  ,     .    S  .   D 

cos  JB  =  cos  — cos f- sin— sin — 

2         2  2        2 

cos^+cos5=     2  cos— cos—  *         (1) 

S'        T) 
cos-4  — cosJB  =  — 2  sin— sin  —  (2) 


EXERCISES.  21 


EXERCISES. 

1.  Find  sine  and  cosine  of  90°  q:  J,  180°:f  ^,  •-.. 

2.  Find  siu3^=  sin(2  J.  +  ^)  =  3  sin^- 4  siuM, 

cos  3  J.  =  cos  {2A-\-A)  =  4:  cos^A  —  3  cos  A. 
3-7.    If  sin  30°  =  |,  (3)  find  sine  and  cosine  of  15°  ; 
(4)  of  45°  ;   (5)  of  22^°  ;   (6)  of  67^°  ; 
(7)  as  answer,  find  those  of  90°,  or  (67^°+ 22^°). 

8.  sina;4-  cosflj=  Vl  +  sin2ic. 

9.  If  Z  tan t  be  read  "the  angle  whose  tangent  is  ^,"  show 
that  Ztan  J  +Ztani  =  45°  ;  also,  ^  +  ^=  90°  if  ^  =  Z  sinf 
and  5  =  Z  sin  |. 

10.  Find  the  area  of  a  regular  dodecagon  inscribed  in  a 
circle  of  radius  12. 

ADDITIONAL. 

1.  If  a  +  6  =  ^-t-^/i,  findsinA 

2.  Find  tan—  by  bisecting  A. 

Hint.  If  the  bisector  divide  sin^  into  two  parts,  x  and  >j,  x:y  = 
1 : cos  A. 

:.  tan  -  =     ■'     =     ^+y     =      sin^ 
2      cos  A     1  +  cos  A      1  +  cos  A 

3.  If  tan-  tan-  tan-  =  1,  find  sin  a  and  cos  a  in  terms  of 

A         '^         Jj 

the  sine  and  cosine  of  h  and  c. 

3'.    If  tan^tan(^45°-:^^cot^=l,    find    the    sine    and 

cosine  of  each  angle  v,  P,  and  E^  in  terms  of  those  of  the 
other  two.     (Due  to  Prof.  O.  Stone.) 

3".    tan  (^45°  _  ^'^  =  Vl  -  sin  P :  Vl  +  sin  P* 


22 


ACADEMIC   TRIGONOMETRY. 


Va 


/  ^''\  /        \ 

&r^    I  I 

.  _i I — J I 


sin.  A 


sin.  B 


COS.  B 


COS.  A 


4.  Find  sum  and  difference  of  sines  and  cosines  directly 
from  figure  and  page  12. 

Hint.  The  diagonals  of  a  rhombus  bisect  at  right  angles.  Half  sum 
=  greater  —  half  difference. 


sin  ^  +  sin  5  =  2  YY^  =:  2  sin  ^  VY'. 


5.  (cos^  -  cos  5)2  +  (sin  ^  -  sin  jB)2=  4  sin^^— -?. 

6.  cos2^  =  (l-tanM)  :  (l  +  tan^^). 

7.  As27rr  =  360%r°  =  — =57.3. 

TT 

How  many  degrees  in  L5r?     Ans. 


270^ 


What  is  the  length  of  80°,  r  being  the  unit? 

Ans.  

57.3 

8.    If  1^=  cos^+  -sin^,  .-.  cos  J.  =   ^  "^^   ~^ ,  and  sin  ^ 


1 


2 

-,  1^  X  Is  being  1(4+5),  and  the  binomial  formula 
z 
holding  for  such  quantity,  find   sin(J.  +  i5),  cos(yL+J5), 
sin  ^  + sin  5,    etc.,    including    cos  2^,    sin  3^,    cos  4yl, 
-f.  =  +V  —  1.     1a=  initial  cos  A  +  transverse  sin  A. 

9.  The  earth's  radius  subtends  an  angle  of  57'  at  the 
moon ;  what  is  the  distance  (by  7)  ?  The  moon's  apparent 
diameter  is  31' ;  what  is  it  in  miles  (by  7)  ? 


EXERCISES.  23 


10.  Construct  the  figure  of  page  13  : 

(1)  When  one  side  is  in  I,  and  the  other  in  II. 

(2)  When  both  are  in  II,  .-.  h  in  I. 

11.  Construct  the  figure  of  page  19  : 

(1)  When  A  and  B  are  in  I,  but  {A  +  B)  in  II. 

(2)  When  A  is  in  I,  and  B  in  II. 

(3)  When  both  are  in  II. 

12.  Find  tan  {A+B)  from  the  figure  of  page  19. 
Hint.   The  base  of  figure  is  now  to  be  taken  as  1. 

Note.    The  "12  additionals  "  are  only  for  the  leaders  of  the  class, 
and  not  for  the  body. 

Part  of  8,  though  sometimes  found  in  Algebra,  seems  more  nearly  in 

place  here. 

\a  =  cos  J.  +  •  sin  Ay  (1) 

I5  ^  cos  J5  +  •  shi  B ;  (2) 

multiplying,  1(^+5)  =  (cos  A  cos  J5  —  sin  J.  sin  B)  +. 

(sin  A  cos  B  -f  cos  A  sin  B), 

but  l(A+5)  =  cos  (^  +  J5)  +  •  sin  {A  +  B). 

.-.  the  usual  formulas. 

If  instead  of  multiplying  (1)  and  (2)  we  add  them,  observing  (p.  22) 

that  a  journey  of  2  cos  —  in  direction  -  causes  the  same  change  of  posi- 

tion  as  the  two  journeys  1^  and  1^,  we  have  the  usual  formulas  for 
sin  ^  +  sin  J5  and  cos  A  +  cos  B. 


^W^" 


24 


ACADEMIC   TllIGONOMETRY. 


PLANE. 

Law  of  Sines.    In  any  plane  triangle,  the  sides  are  propor- 
tional to  the  sines  of  the  opposite  angles. 


.  a  :  6  =  sin  ^  :  sin  B. 


By  definition  of  sine, 

a  sin  JB  =  J)  =  6  sin  A. 

Law  of  Cosines.    The  square  of  any  side  of  a  (pi.)  triangle 
is  equal  to  the  sum  of  the  squares  of  the  other  two  sides, 
minus  twice  their  product   by  the   cosine   of  the  included 
angle. 
Pythagorean  Proposition  : 

a^  =  p^  -\-m^,  b^=p^-\-n^. 
.*.  a^  —  h^  =  iw  —  n^.     (In  words.) 
a^-b''  =  {c-7iy-ri'  =  c''-2cn. 

Definition  of  Cosine  : 

nz=b  cos  A.  - 

,'.  substituting  and  transposing  term  5^, 

a^=b'--\-c'-2bccosA. 

Law  of  Tangents.  The  sum  of  any  two  sides  of  a  (pi.)  tri- 
angle is  to  their  difference  as  the  tangent  of  one-half  the  sum 
of  the  opposite  angles  is  to  the  tangent  of  one-half  their 
difference.  ' 

By  law  of  sines  and  theory  of  proportion, 

ft  +  &  _  sin  ^  +  si"  I^  ^  tan  |^(^1  -f-  B) 
a  —  b      sin^  — sin5      t^ni{A—  B) 


LAWS.  25 


SPHERICAL. 

Law  of  Sines.    In  any  spherical  triangle,  the  sines  of  the 
sides  are  proportional  to  the  sines  of  the  opposite  angles. 


^    71  X)   m        D^--n^ 

By  sin  Ay., 

sin  asmB=  sinp  =  sin  b  sin  A. 
.'.  sin  a  :  sin  b  =  sin  A  :  sin  B. 

Law  of  Cosines.  The  cosine  of  any  side  of  a  (sph.)  triangle 
is  equal  to  the  product  of  the  cosines  of  the  other  two  sides, 
plus  the  product  of  their  sines  by  the  cosine  of  their  included 
angle. 

Pythagorean  Analogy  : 

cos  a  =  Qosp  cos  m,  cos  b  —  cosp  cos  n. 
.'.  cos  a  :  cos6  =  cosm  :  cos?i.     (In  words.) 
cosa  :  cos6  =  cos(c— ??):  eos?i 

=  cosc-f-  sine  tan 7i. 
Cosine  Analogy : 

tann  =  tan  6  cos^. 

Substituting  and  transposing /ac^or  cosb, 

cos «  =  cos  &  cose  +  sin  6  sine  cos ^.  % 

For  Analogy  of  Law  of  Tangents,  see  the  limit  of  Napier's 
(2),  page  29. 


26  ACADEMIC   TRIGONOMETHY. 

FUNCTIONS  OF   HALF  ANGLES. 

Plane.    By  law  of  cosines, 

cos  A  = ^^ — - — ^  =  —^ 

-26c  26c 

Problem.    To  substitute  this  value  in  licosJ^,  and  thus 
find  the  functions  of  the  half  angles  in  terms  of  the  sides. 
If  a4-6  +  c=2s, 

.'.  a  +  6  —  c=  2(s  — c), 
a— 6  +  c=2(s  — 6), 
6  +  c  —  a=  2(s  —  a). 

1— cos^  = ' = ^^ -^ 

26c  26c 

^2(^-6)2(g-c) 
26c 

Similarly,  1  +  cos  ^  =  2s{s-a)^ 

be 


sm 


A        l(s-b)(s-c)      -          .A        ll-cos^ 
—  =A  -^^ — -]  from  sm—  =\ . 

2       \  6c  2       \        2 


A        Is(s-a)  .  .4        |l  + 

—  =  \  -^^ ^:  from  cos -  =  -\ — - 

2       \       6c       '  2       \ 


cos^ 


COS 

tan^^JI^-^)(--^). 
2       \      s{s-a) 

That  is,  the  sine  of  half  an  angle  of  a  plane  triangle  is 
equal  to  the  square  root  of  half  the  sum  of  the  three  sides 
minus  one  of  the  including  sides,  into  the  half  sum  minus  the 
other  including  side,  divided  by  the  product  of  the  including 
sides. 

Give  for  cosine  and  tangent. 

Ex.  Find  the  value  of  tan—  directly  from  page  12,  and 
Wentworth's  Geometry,  page  250. 


HALF   ANGLES.  27 


If  r  be  the  radius  of  inscribed  circle,  area  =  rs. 
By  Geometry, 
an 


-ea  =  Vs(s  —  a)  (s  —  h)  (s  —  c) , 

^,^     \(s-a){s-h){s-c) 
\  s 


By  figure,  tan— = 


2      s  —  a 

FORMULAS  FOR   HALF   ANGLES. 

Spherical.    By  law  of  cosines, 

.      cos  a  —  cos  h  cos  c 

cos  A  = • 

sin  b  sin  c 

As  in  Plane, 

sin  b  sin  c  +  cos  6  cos  c  —  cos  a 


1  —  cos  A  = 


sin  b  sin  c 


cosa  — cos(6  — c)      /  V*     i?        \ 
= V 1      (clif .  of  cos) 

—  sin  b  sin  c 
_  2  sin(g  —  6)  sin (8  —  c) 
sin 6  sine 
Similarly, 

sin 6  sine 
.-.  the  analogy  here  is  the  same  as  in  law  of  sines, 


A_    j8in(s  — 


sm__    ,      ^   -fe)sin(s-c). 


sin  b  sin  c 


^  _     Isin  s 


^^„—        , sin(s  — a) 

cos  —  —     ' ^^ ^' 


sin  b  sin  c 


tan-=    |sin(.s-6)sin(s-c) 
2      "V      sinssin(s  — a) 

That  is,  the  cosine  of  one-half  of  either  angle  of  a  spheri- 
cal triangle  is  equal  to  the  square  root  of  the  sine  of  one-half 
of  the  sum  of  the  three  sides,  into  the  sine  of  one-half  this 


28  ACABEIkUC  TRIGONOMETRY. 

sum  minus  the  side  opposite  the  angle,  divided  by  the  product 
of  the  sines  of  the  including  sides. 

Giv'e  sine  and  tangent  in  words.  ^ 

LAW  OF  COSINES  FOR  ANGLES.    FUNCTIONS   OF 
HALF   SIDES. 

If  A'B'C  be  the  polar  of  ABC,  by  first  law  of  cosines  and 

Geometry, 

cos^  =  —  cos  jB  cos  (7+  sin^sinCcosa. 
In  words, 

cos  A  4-  cos  B  cos  G 

cosa  = /       .      , 

sin  i5  sm  G 


H 


—  GosS  cos{S  —  A) 


sin 

sin  B  sin  C 


cos 


2=\ 


-  B)  cosjS  -  C) 


sin  B  sin  C 
In  words.     Note  Analogies. 

THE   GAUSS   EQUATIONS. 


Bin  i(A-\-B)  cOs^c  =  cos^{a  —  b)  cos^C. 

cos^{A  +  B)  Gos^c  =  cos|(a  +  b)  sin^C. 

sin  i{A  —  B)  sin^c  =  sin^(a  —  b)  gos^O. 

cosi{A  —  B)  s'm^c  =  sin  ^(a  +  5)  sin  ^  C. 

fH^±B)     flc=    /i(a±6)    no. 

Rule  I.     sin  in  (1)  gives  —  in  (3)  and  conversely. 

cos  in  (1)  gives  -f  in  (3)  and  conversely. 

Rule  II.   Functions  have  same  names  in  (2)  and  (3). 

Functions  have  co-names  in  (4)  and  (1). 
Note.   The  rules  also  hold  for  I,  II,  III,  and  IV. 


napier's  pkoportions.  29 


To  prove  (1),  smi{A  -h B)=  smf^  +  ^\ 

Now  sin  (  —  H — )  =  sin—  cos  — h  cos  — sin  — 

\2       2j  2         2  2        2 


_     /siu(s  —  6)  sin(s  —  c)     /sing  sin(8  — &) 
\  sin  &  sin  c  \      sin  a  sin  c 

/sin  g  sin  (s  —  a)     /sin(s  —  a)  sin  (.9  —  c) 
\      sin  b  sin  c        \  sin  a  sin  c 

sin(s  — 6)        (7  ,  sin(s  — a)        C 
=  — ^^ ^cos 1 ^^ ^cos- 
sin  c             2            sine  2 

sin(s  —  6)-fsin(s  — a)        C     .^  ^    -       \ 

=  — \ L^ ^ ^cos—     (bv  sum  of  sines) 

sine  2     ^  -  . 

2  sinf  s ^^!^— icos 


C 
cos  — ; 


sine  2 


u  4.  a  -\-h       c 

but  s -^^-  =  — 

2  2 


2  sin  —  cos  — 


C 

cos .*.  Q.  E.  D. 


£i     •      C  C  2  . 

2  Sin— COS—  ^ 

2         2 

By  the  Gauss  Equations  and  (Div.  Ax.)  we  have 

NAPIER'S  PROPORTIONS. 

(1)  sini(a  +  b):sini(a  -  b)  :  icotiC :  tani(A- B). 

(2)  cosi(a  +  b):cosi(a  -  b)  : :  cot^C :  tani^A  -\-  B) . 

(3)  sin^(^-f-5):sin^(yl-5)::tanic  :  tan^Ca  -  b). 

(4)  cosi(^  +  J5):cosi(^-S)::tan^c  :tan^(a  +  b). 

Theorem  1 .  The  sine  of  one-half  the  sum  of  either  tivo  sides 
of  any  spherical  triangle  is  to  the  sine  of  one-half  their  differ- 
ence as  the  cotangent  of  one-half  the  angle  ichich  they  include 
is  to  the  tangent  of  one-half  the  difference  of  the  angles  opposite. 


30 


ACADEMIC    TKIGONOMETRY. 


Theorem  2.  The  cosine  of  one-half  the  sum  of  either  two 
sides  of  any  spherical  triangle  is  to  the  cosine  of  one-half  their 
difference  as  the  cotangent  of  07ie-half  the  angle  ivhich  they 
include  is  to  the  tangeyit  of  one-half  the  sum  of  the  angles 
opposite. 

Theorem  3.  The  sine  of  one-half  the  sum  of  either  two  angles 
of  any  spherical  triangle  is  to  the  sine  of  one-half  their  differ- 
ence as  the  tangent  of  one-half  the  side  which  they  include  is  to 
the  tangent  of  one-half  the  difference  of  the  sides  opposite. 

Theorem  4.  The  cosine  of  one-half  the  sum  of  either  two 
angles  of  any  spherical  triangle  is  to  the  cosine  of  one-half  their 
difference  as  the  tangent  of  one-half  the  side  ichich  they  include 
is  to  the  tangent  of  one-half  the  sum  of  the  sides  opposite. 

Note  the  analogies:  sine  giving  — ,  cosine,  +  ;  side  giv- 
ing "co"/,  angle  giving  function. 

EXAMPLES  UNDER  OBLIQUE  TRIANGLES. 
Given  (any  three  parts,  one  being  a  side)  : 

I.  One  side  and  two  angles,  L.  of  Ss. 
II.  Two  sides  and  Z  opposite  one  of  them,  L.  of  Ss. 

III.  Two  sides  and  the  included  Z,  L.  of  Ts.  or  L.  of  Cs. 

IV.  Three  sides.     Formulas  for  half  A  or  L.  of  Cs. 
V.  Sph.    Three  A.     Formulas  for  half  sides. 

I.   {a,A,B)  (b,A,B)  (c,A,B)  (a,A,C)  (b,A,C) 

(c,A,C)  {a,B,C)  (h,B,C)  (c,B,C). 
II.   (a,b,A)  {a,b,B)  {a,c,A)  {a,c,C)  (b,c,B)  {b,c,C). 
III.    (a,  6,0)  {a,c,B)  {b,c,A). 


a 

b 

c 

A 

B 

C 

3. 

1686 

960 

2400 

33°  34'  39" 

18°  21'  21" 

128°  4' 

4. 

40 

34 

45 

83°  53'  15" 

57°  41'  25" 

38°  25'  20" 

5. 

10 

12 

14 

44°  24'  65" 

57°  7' 18" 

78° 27' 47" 

1. 

6 

8 

10 

36°  52' 

53°  8' 

90° 

2. 

2 

Vo 

45^ 

60° 

75° 

ASTRONOMICAL    TRIANGLES. 


31 


6.  Find  the  distance  between  two  objects  (supposed  inac- 
cessible) by  calculating  the  distance  to  each  by  right  triangles, 
also  by  law  of  sines,  then  measuring  the  angle  between  these 
distances.     Measure  the  distance  to  test  your  answer. 

7.  If  in  (4)  the  unit  be  one  mile,  and  CA  be  east  and 
west  line,  what  is  the  direction  of  each  vertex  from  the 
others  ? 

8.  An  object  when  viewed  from  the  ends  of  an  east  and 
west  line,  of  length  34,  bears  N.  51°  35'  W.  and  N.  6°  7'  E. 
What  is  its  distance  from  each,  and  from  the  straight  line 
joining  them  ? 


a 

b 

c 

A 

B 

C 

1. 

2. 
3. 

4. 

70O  4' 17" 
20O16' 
147°  5' 33" 

69°  34' 56" 

63°  21' 26" 
56°  18' 
165°  5' 18" 

58^16' 22" 
66°  18' 
33°  1'36" 
70°  20' 20" 

81°  38' 20" 
20°  10' 
110°  10' 
50°  10' 10" 

70°  9' 38" 
550  54' 
133°  18' 

63°  31' 20" 

114°  18' 30" 

70°  20' 40" 

50^30'  8" 

As  the  chief  applications  are  to  Astronomy,  we  give 


THE  FIRST  ASTRONOMICAL  TRIANGLE  PZH. 

P  is  north  pole  of  equator. 

Z  is  zenith  (pole  of  horizon) . 

IT  is  the  heavenly  body. 

Altitude   of  pole  =  AT' =  latitude   of  N\ 
place. 

The  distances  of  a  body  from  any  great 
circle  and  its  pole  are  complementary. 

PZ  =  co-latitude.    HP—  polar  distance  =  co-declination. 

HZ=  zenith  distance  =  co-altitude. 

Z. Z=  azimuth.    AP—  hour  or  time  angle. 

Example.   Calculate  the  following  for  a  place,  latitude  45°, 
and  for  the  longest  day  of  the  year. 


32 


ACADEMIC    TKIGONOMETllY. 


(a)  The  azimuth  of  sun  at  setting  and  rising. 

(b)  Time  of  rising  and  setting. 

(c)  The  greatest  altitude  of  the  sun. 

(d)  Time  when  a  vertical  object  casts  a  horizontal  shadow 
of  its  own  length. 


THE   SECOND  ASTRONOMICAL  TRIANGLE. 

Before  considering  this  triangle  involving  two  great  circles 
and  their  poles,  we  will  consider  a  point  referred  to  any  great 
circle  and  its  pole. 

Let  F  (any  point  of  this  great  circle)  be  the  origin,  and 
consider  the  hemisphere  limited  by  the  great  circle  of  F. 


P  b 


P  h 


E 


E 


f^ 

^ 

\ 

^ 

/^A^ 

\ 

M 

\    '     In 

V    ^ 

^ 

\/ 1 

yj.'AQ 

\x 

/W^ 

■^  1 

V 

v^ 

V 

Let  P  be  the  pole,  and  EQ,  the  great  circle  of  reference, 
H  the  heavenly  body  ;  VH  is,  by  Geometry,  _L  to  great  circle 
of  F  cutting  it  in  (r ;  denote  PQ  by  5,  HG  by  a. 

VF^:^  dii-ect  co-ordinate  D,  FQ  =  co.  D.     (Fig.  2.) 
FH=  transverse  co-ordinate  T,  HP  =co.  T. 
From  the  rt.  A  HPG  and  HVF  (C.  Ay.), 
tan  h  —  cot  T  sin  Z). 
.*.  tan  r  =  cot  &  sin  Z). 

tana 


(Tan.  Ay.),    tan(co.  i>)    =  cotD 
(P.  Ay.) ,        cos  DcosT=  cos  VH. 


sin  6 


(1) 

(2) 
(3) 
(4) 


In  the  second  triangle  PP'H,  we  have  two  systems  of  co- 
ordinates :  Right  ascension  {R)  and  declination  (d),  when 


REDUCTION   OP   OBSERVATION. 


33 


referred  to  the  equator  EQ^  but  latitude  I  and  longitude  L^ 
when  referred  to  the  ecliptic  EG.  Since,  if  the  former  be  D 
and  T,  the  latter  are  Z)'  and  T',  if  F,  the  common  point  of 
the  two  circles,  be  the  origin : 

tan  h  =  cot  d  sin  R. 
sin  6  _  tan  R 
sin  6' 


•         (1) 

(2)  by  (3)  above, 
taniv 

tan  6' =  cot  Z  sin  iv.  (3) 

cos  I  cos  L  =  cos  d  cos  R.  (4) 

PF=  E,  the  obliquity  of  the  ecliptic  ;  P'  =  co.i  ;  P=90+i2. 

The  reduction  of  an  observation  from  the  equator  to  the 
ecliptic  is  of  so  much  importance  that  one  example  is  worked 


through  and  back. 

M 

81°  48'  42.4" 

L 

85°  45'  2.00" 

d 

68°  27'  54.5" 

I 

45°    7' 48.98" 

sinU 

9.9955499    " 

sinL 

9.9988044 

coid 

9.5961411 

cot  I 

9.9980251 

tan  b^ 

9.5916910 

tan  6 

9.9968295 

h' 

21°  20'  1.58" 

b 

44°  47'  27.11" 

e 

23°  27'  25.53" 

e 

23°  27'  25.53" 

b=b'-\-e 

44°  47'  27.11" 

b'=b-e 

21°  20'  1.58" 

sin  b 

9.8478940 

sin  b^ 

9.5608631 

1 :  sin  b^ 

0.4391369 

1 :  sin  6 

0.1521060 

tanU 

0.8419622 

tan  L 

1.1289931 

tanL 

1.1289931 

tanjR 

0.8419622 

L 

85°  45'  2.00" 

JB 

81°  48'  42.4" 

sin  L 

9.9988044 

sin  JB 

9.9955499 

cot  6 

0.0031705 

cot  b' 

0.4083090 

tan  I 

0.0019749 

iand 

0.4038589 

I 

45°  V  48.98" 

d 

68°  27'  54.5" 

cos  I 

9.8484953 

cose? 

9.5647190 

COS  L 

9.8698113 

cos  It 

9.1535877 

COS  /  COS  L 

8.7183066 

COS  dcosR 

8.7183067 

34  ACADEMIC   TRIGONOMETRY. 

QUADRANTAL  TRIANGLES. 

By  Geometry,  the  polar  of  a  right  triangle  is  a  quadrantal 
triangle.  If  A'B'C  be  a  rt.  A,  and  ABC  its  polar,  then,  by 
Napier's'Bules, 

sin^=      tan  ^  cot  &  =      sin  a  sin^. 

sin  B  —      tan  A  cot  a  =      sin  6  sin  H. 

cos  a  =  —  tan  B  cot  H=     cos  A  sin  h. 

cos  6  =  —  tan^  cot  jEr=      cos  J5  sin  a. 

cos  H—  —  cot  a  cot  5  =  —  cos  A  cos  ^. 

That  is,  Napier's  Rules  would  hold  for  a  quadrantal  tri- 
angle, if  the  circular  parts  be  the  two  angles  about  the 
quadrant,  the  complements  of  the  opposite  sides,  and  the 
complement  of  the  hypotenuse  angle,  giving 

THREE   SECONDARY  ANALOGIES. 

sin^  tan-B       .  tan^ 

sm  a  = •>   cos  a  = ?    tan  a  —  — - —  • 

smlT  tan^  sin  ^ 

Rule.  Those  formulas  that  contain  a  co-function  of  H 
have  a  negative  sign. 

I.    The  same  as  for  right  triangles. 

Proof  by  I,  of  rt.  A,  if  ^^90%  a'^OO^,  ^'^90°,  .-.  a^90°. ' 

II'.    The  reverse  of  II,  for  right  triangles. 

Proof.  If  A  and  B  in  same  quadrant,  .-.a'  and  6'  in 
same. 

By  II,  /i'<  90%  .-.  H>  90°.  q.  e.  d. 

Proof  of  Rule.  There  are  but  two  hypotheses  : 
(a)  A  and  B  in  the  same  quadrant ; 
(6)  A  and  B  in  different  quadrants. 

If  (a),  by  II",  H>  90°,  .-.  co-function  H  -. 
But,  by  I',  the  other  two  factors  are  alike  in  signs. 
,-.  sign  of  formula  must  be  —. 


QUADRANTAL   TKIANGLES.  3t5 

If  (6),  by  II,  H<  90°,  .-.  co-function  of  //  -|-. 
But  by  I,  the  other  two  factors  are  alike  in  sign. 
.-.  sign  of  formula  must  be  — .     .*.  q.  e.  d. 

EXAMPLES   UNDER  QUADRANTAL  TRIANGLES. 

1.  In  what  latitude  does  the  sun  rise  in  the  northeast  at 
Glimmer  solstice  ? 

2.  Find  time  of  sunrise  and  sunset  at  your  school  on  the 
longest  day  of  the  year. 

3.  Rt.  A.  Form  a  table  of  times  when  the  shadow  of  the 
side  of  the  east  school  door-way  will  coincide  with  an  east 
and  west  crack  in  the  hall  floor. 

Closing  Note.  Establish  the  definitions  of  sine  and  cosine  without 
similar  triangles,  thence  laws  of  sine  and  cosine  and  Pythagorean 
proposition.  Then  give  trigonometric  proofs  of  proportions  concerning 
oblique  lines,  sides  of  a  triangle  opposite  equal  angles,  and  converse. 

Greater  side  opposite  greater  angle,  and  converse. 

Three  propositions  concerning  similar  triangles,  and  three  concern- 
ing equal  triangles,  also  any  other  trigonometric  proofs  of  common 
geometric  principles,  thus  tieing  the  two  subjects  together. 


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68  MATHEMATICS. 

Wentworth's  Trigonometries. 

TIHE  aim  has  been  to  furnish  just  so  much  of  Trigonometry  aa 
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selected  with  a  view  to  awaken  a  real  love  for  the  study.  Much 
time  and  labor  have  been  spent  in  devising  the  simplest  proofs  for 
the  propositions,  and  in  exhibiting  the  best  methods  of  arranging 
the  logarithmic  work.     Answers  are  included. 

Plane  and  Solid  Geometry,  and  Plane  Trigonometry. 

12mo.    Half  morocco.    490  pages.    Mailing  Price,  |1.55;  Introduction, 
f  1.40;  Allowance  for  old  book,  40  cents. 

Plane  Trigonometry. 

12mo.    Paper.  80  pages.    Mailing  Price,  35  cents ;  Introduction,  30  cents. 

Plane  Trigonometry  Formulas. 

Two  charts  (30  x  40  inches  each)  for  hanging  on  the  walls  of  the  class- 
room.   Introduction  Price,  ;i?l-00  per  set. 

Plane  Trigonometry  and  Logarithms. 
8vo.    Paper.   160  pages.    Mailing  Price,  G5  cents ;  Introduction,  60  cents. 

Plane  and  Spherical  Trigonometry. 

12mo.     Half  morocco,     iv  + 151  pages.     Mailing  Price,  80  cents ;  for 
introduction,  75  cents. 

Wentworth's  Plane  and  Spherical  Trigonometry, 

and  Surveying. 

With  Tables.    8vo.    Half  morocco.    307  pages.    Mailing  Price,  $1.40; 
Introduction,  $1.25;  Allowance  for  old  book,  40  cents. 

Surveying. 

8vo.    80  pages.    Paper.    Mailing  Price,  35  cents  ;  for  introduction,  3C 
cents. 

Wentworth's  Plane  and  Spherical  Trigonometry, 

Surveying,  and  Navigation. 

12mo.    Half  morocco.    330  pages.    Mailing  Price,  $1.25;  Introduction, 
$1.12;  Allowance  for  old  book,  40  cents. 


MATHEMATICS. 


69 


rpHE  object  of  the  work  on  Surveying  and  Navigation  is  to  pre- 
sent  these  subjects  in  a  clear  and  intelligible  way,  according 
to  the  best  methods  in  actual  use ;  and  also  to  present  them  in  so 
small  a  compass,  that  students  in  general  may  find  the  time  to 
acquire  a  competent  knowledge  of  these  very  interesting  and 
important  studies.     Answers  are  included. 


S.  J.  Kirkwood,  Prof,  of  Mathe- 
matics, University  of  Wooster,  O.: 
Wentworth's  Algebra,  Geometry, 
and  Trigonometry  are  excellent  text- 
books.   {Dec.  15,  1883.) 


Otis  H.  Robinson,  Prof,  of  Mathe- 
matics, University  of  Rochester :  I 
think  Wentworth's  Surveying  an 
admirable  introduction  to  the  study 
of  the  subject.    {May  28,  1883.) 


Wentworth  &  Hill's  Fiue-Place  Logarithmic  and 

Trigonometric  Tables. 

By  G.  A.  Wentworth,  A.M.,  and  G.  A.  Hill,  A.M. 

Seven  Tables  (for  Trigonometry  and  Surveying) :  Cloth.   8vo.    79  pages. 
Mailing  Price,  55  cents;  Introduction,  50  cents. 

Complete  (for  Trigonometry,  Surveying,  and  Navigation) :   Half  mo- 
rocco.   8vo.    158  pages.    Mailing  Price,  $1.10;  Introduction,  $1.00. 

rpHESE  Tables  have  been  prepared  mainly  from  Gauss's  Tables, 
and  are  designed  for  the  use  of  schools  and  colleges.  They 
are  preceded  by  an  introduction,  in  which  the  nature  and  use  of 
logarithms  are  explained,  and  all  necessary  instruction  given  for 
using  the  tables.  They  are  printed  in  large  type  with  very  open 
spacing.  Compactness,  simple  arrangement,  and  figures  large 
enough  not  to  strain  the  eyes,  are  among  the  points  in  their  favor. 


Wentworth  &  Hill's  Exercises  in  Arithmetic. 

I.  Exercise  Manual.  12mo.  Boards :  vi  +  282  pages.  Mailing  Price, 
55  cents;  for  introduction,  50  cents.  11.  Examination  Manual.  12mo. 
Boards.  148  pages.  Mailing  Price,  40  cents ;  Introduction  Price,  35  cents. 
Both  in  one  volume,  80  cents.    Answers  to  both  parts  together,  10  cents. 

nnilE  first  part  (Exercise  Manual)  contains  3869  examples  and 
problems  for  daily  practice,  classified   and  arranged  in  the 
common  order;  and  the  second  part  (Examination  Manual)  con- 
tains 300  examination-papers,  progressive  in  character. 


70  MATHEMATICS. 

Wentworth  &  Hill's  Exercises  in  Algebra. 

I.  Exercise  Manual.  12mo.  Boards.  232  pages.  Mailing  Price,  40 
cents;  Introduction  Price,  35  cents.  II.  Examination  Manual.  12mo. 
Boards.  159  pages.  Mailing  Price,  40  cents;  Introduction  Price,  35 
cents.  Both  in  one  volume,  70  cents.  Answers  to  both  parts  together, 
25  cents. 


T 


HE  first  part  (Exercise  Manual)  contains  about  4500  problems 
classified  and  arranged  according  to  the  usual  order  of  text- 
books in  Algebra;  and  the  second  part  (Examination  Manual) 
contains  nearly  300  examination-papers,  progressive  in  charac'. 
ter,  and  well  adapted  to  cultivate  skill  and  rapidity  in  solving 
problems. 

British  Mail:  All  engaged  in  the 
practical  work  of  education  will 
appreciate  tliese  Manuals,  as  they 
are  calculated  to  save  the  master 


much  precious  time  and  labor,  and 
to  give  his  students  the  benefit  of 
progressive  and  carefully  thought- 
out  exercises. 


Wentworth  &  Hill's  Exercises  in  Geometry. 

12mo.    Cloth.    255  pages.    Mailing  Price,  80  cents ;  Introduction  Price, 
70  cents. 

nPHE  exercises  consist  of  a  great  number  of  easy  problems  for 
beginners,  and  enough  harder  ones  for  more  advanced  pupils. 
The  problems  of  each  section  are  carefully  graded,  and  some  of  the 
more  difficult  sections  can  be  omitted  without  destroying  the  unity 
of  the  work.  The  book  can  be  used  in  connection  with  any  text- 
book on  Geometry  as  soon  as  the  geometrical  processes  of  reason- 
ing are  well  understood. 


select  propositions  from  it  to  supple- 
ment every  stage  of  our  work. 


Amelia  W.  Platter,  Hiyh  School, 
Indianapolis,  Ind. :  I  find  the  sub- 
ject so  carefully  graded,  that  I  can 

Analytic  Geometry. 

By  G.  A.  Wentworth.  Revised  edition.  12mo.  Half  morocco,  xii  + 
301  pages.  Mailing  Price,  $1.35;  for  introduction,  $1.25;  allowance  in 
exchange,  30  cents. 

rpiiE  aim  of  this  work  is  to  present  the  elementary  parts  of  the 
subject  in  the  best  form  for  class-room  use. 
The  connection  between  a  locus  and  its  equation  is  made  per- 
fectly clear  in  the  opening  chapter. 


MATHEMATICS. 


71 


The  exercises  are  well  graded  and  desired  to  secure  the  best 
mental  training. 

By  adding  a  supplement  to  each  chapter  provision  is  made  for  a 
shorter  or  more  extended  course,  as  the  time  given  to  the  subject 
will  permit. 

The  book  is  divided  into  chapters  as  follows  :  — 

PART  I.    Plane  Geometry.      I.    Loci  and  their  Equations; 

II.  The  Straight  Line;  IIL  The  Circle;  IV.  Different  Systems  of 
Co-ordinates ;  V.  The  Parabola ;  VI.  The  Ellipse ;  VII.  The  Hyper- 
bola; VIIL  Loci  of  the  Second  Order;  IX.  Higher  Plane  Curves. 

PART  II.    Solid  Geometry.     I.  The  Point;  II.  The  Plane; 

III.  The  Straight  Line ;  IV.  Surfaces  of  Revolution. 


Dascom  Greene,  Prof,  of  Mathe- 
matics and  Astronomy,  Rensselaer 
Polytechnic  Institute,  Troy,  N.Y.  : 
It  appears  to  be  admirably  adapted 
to  tbe  use  of  beginners,  and  is  espe- 
cially rich  in  examples  for  practical 
application  of  the  principles  of  each 
chapter.  The  full  and  clear  explana- 
tion of  first  principles  given  in  the 
opening  chapter  is  a  new  and  highly 
commendable  feature  of  the  work. 
{Nov.  11,  1886.) 

Geo.  D.  Olds,  Prof,  of  Mathematics, 
University  of  Rochester,  N.  Y. :  It  is 
a  most  admirable  little  book.  The 
author  falls  into  line  with  what  I 
believe  to  be  the  best  modern  ten- 
dency in  text-books,  —  the  avoidance 
of  bulk  and  complexity. 
(JVo?J.  11,  1886.) 

J.  L.  Patterson,  Teacher  of  Mathe- 
matics, Laivrenceville  School,  2i.J. : 
I  do  not  know  of  any  text-book  for 
beginners  in  this  subject  which  can 
compare  with  it  for  class-room  use. 

E.  A.  Paul,  Prin.  of  High  School, 
Washington,  D.C. :  I  think  it  is  to  be 
commended  for  the  same  clearness 
of  statement  and  simplicity  of  ar- 


rangement for  which  the  author's 
other  works  are  noted,  and  believe 
it  to  be  especially  adapted  for  ad- 
vanced pupils,  in  high  schools  and 
academies,  who  wish  to  know  some- 
thing of  the  mysteries  of  loci  and 
conic  sections,  and  who  have  only  a 
limited  time  for  the  work. 
{Nov.  12,  1886.) 

Jos.  J.  Hardy,  Prof,  of  Mathe- 
matics, Lafayette  College,  Easton, 
Pa. :  The  professor's  experience  has 
taught  him  what  are  the  points  which 
the  boys  find  obscure,  and  he  has 
generally  been  successful  in  devising 
a  good  way  of  elucidating  these 
ix)ints.  .  .  .  Teachers  will  find  it  a 
very  helpful  manual. 
{Jan.  4, 1887.) 

E.  Miller,  Prof,  of  Mathematics, 
University  of  Kansas,  Laivrence : 
As  a  book  for  beginners,  it  is  admi- 
rable in  all  its  arrangements  and 
features.  The  great  number  of 
problems  scattered  through  it  will 
largely  relieve  it  of  that  abstract 
analysis  which  is  so  often  a  terror  to 
students.  The  book  is,  like  the  other 
works  of  Professor  Wentworth,  a 
good  thing.    (Nov.  18,  1886.) 


72  MATHEMATICS. 

A  Treatise  on  Plane  Surueying. 

By  Daniel  Cabhart,  C.E.,  Professor  of  Civil  Engineering  in  the  West- 
ern University  of  Pennsylvania,  Allegheny.  Illustrated.  8vo.  Half 
leather,  xvii  +  498  pages.    Mailing  Price,  $2.00;  for  introduction,  $1.80. 

rpmS  work  covers  the  whole  ground  of  Plane  Surveying.  It 
illustrates  and  describes  the  instruments  employed,  their  ad- 
justments and  uses;  it  exemplifies  the  best  methods  of  solving  the 
ordinary  problems  occurring  in  practice,  and  furnishes  solutions 
for  many  special  cases  which  not  infrequently  present  themselves. 
It  is  the  result  of  twenty  years'  experience  in  the  field  and  technical 
schools,  and  the  aim  has  been  to  make  it  extremely  practical,  having 
in  mind  always  that  to  become  a  reliable  surveyor  the  student  needs 
frequently  to  manipulate  the  various  surveying  instruments  in  the 
field,  to  solve  many  examples  in  the  class-room,  and  ,to  exercise 
good  judgment  in  all  these  operations.  Not  only,  therefore,  are 
the  different  methods  of  surveying  treated,  and  directions  for  using 
the  instruments  given,  but  these  are  supplemented  by  various  field 
exercises  to  be  performed,  by  numerous  examples  to  be  wrought, 
and  by  many  queries  to  be  answered. 

Chapter  I.   Chain  Surveying. 

"       II.   Compass  and  Transit  Surveying. 

"      III.  Declination  of  the  Needle. 

"      IV.   Laying  Out  and  Dividing  Land. 

"        V.   Plane  Table  Surveying. 

"      VI.   Government  Surveying. 

"     VII.   City  Surveying.     Including  the  Principles  of  Levelling. 

"  VIII.   Mine  Survejring.    Including  the  Elements  of  Topography. 

The  following  Tables  have  been  added:  Logarithms  of  num- 
bers ;  Approximate  equation  of  time ;  Logarithms  of  trigonometric 
functions ;  For  determining  with  greater  accuracy  than  the  pre- 
ceding ;  Lengths  of  degrees  of  latitude  and  longitude ;  Miscellaneous 
formulas,  and  equivalents  of  metres,  chains,  and  feet;  Traverse; 
Natural  sines  and  cosines ;  Natural  tangents  and  cotangents. 

The  judicial  functions  of  surveyors,  as  given  by  Chief  Justice 
Cooley,  are  set  forth  in  an  appendix. 

The  work  is  published  just  as  this  Catalogue  goes  to  press,  so 
that  full  notices  cannot  be  given.     Send  for  the  special  circular. 


MATHEMATICS.  77 

Byerly^s  Syllabi. 

By  W.  E.  Byerly,  Professor  of  Mathematics  in  Harvard  University. 
Each,  8  or  12  pages,  10  cents. 

Syllabus  of  a  Course  in  Plane  Trigonometry. 

Syllabus  of  a  Course  in  Plane  Analytical  Geometry. 

Syllabus  of  a  Course  in  Plane  Analytic  Geometry.   (Advanced  Course.) 

Syllabus  of  a  Course  in  Analytical  Geometry  of  Three  Dimensions. 

Syllabus  of  a  Course  on  Modern  Methods  in  Analytic  Geometry. 

Syllabus  of  a  Course  in  the  Theory  of  Equations. 

Elements  of  the  Differential  and  Integral  Calculus. 

With  Examples  and  Applications.  By  J.  M.  Taylor,  Professor  of 
Mathematics  in  Madison  University.  8vo.  Cloth.  249  pages.  Mailing 
Price,  $1.95;  Introduction  Price,  $1.80. 

rpHE  aim  of  this  treatise  is  to  present  simply  and  concisely  the 
fundamental  problems  of  the  Calculus,  their  solution,  and 
more  common  applications.  Its  axiomatic  datum  is  that  the 
change  of  a  variable,  when  not  uniform,  may  be  conceived  as 
becoming  uniform  at  any  value  of  the  variable. 

It  employs  the  conception  of  rates,  which  aifords  finite  differen- 
tials, and  also  the  simplest  and  most  natural  view  of  the  problem  of 
the  Differential  Calculus.  This  problem  of  Jinding  the  relative 
rates  of  change  of  related  variables  is  afterwards  reduced  to  that  of 
finding  the  limit  of  the  ratio  of  their  simultaneous  increments  ; 
and,  in  a  final  chapter,  the  latter  problem  is  solved  by  the  principles 
of  infinitesimals. 

Many  theorems  are  proved  both  by  the  method  of  rates  and  that 
of  limits,  and  thus  each  is  made  to  throw  light  upon  the  other. 
The  chapter  on  differentiation  is  followed  by  one  on  direct  integra- 
tion and  its  more  important  applications.  Throughout  the  work 
there  are  numerous  practical  problems  in  Geometry  and  Mechanics, 
which  serve  to  exhibit  the  power  and  use  of  the  science,  and  to 
excite  and  keep  alive  the  interest  of  the  student. 


78 


MATHEMATICS. 


The  Nation,  New  York:  It  has 
two  marked  characteristics.  In  the 
first  place,  it  is  evidently  a  most 
carefully  written  book.  There  is 
nothing  vague  or  slipshod  in  it. 
Nearly  every  sentence,  certainly 
every  theorem,  seems  to  have  heen 
constructed  with  a  strenuous  effort 
to  give  it  clearness  and  precision. 
This  constant  attention  to  the  form 
of  expression  has  enabled  the  author 
to  be  concise  without  becoming  ob- 
scure. We  are  acquainted  with  no 
text-book  of  the  calculus  which  com- 
presses so  much  matter  into  so  few 


pages,  and  at  the  same  time  leaves 
the  impression  that  all  that  is  neces- 
sary has  been  said.  In  the  second 
place,  the  number  of  carefully  se- 
lected examples,  both  of  those  worked 
out  in  full  in  illustration  of  the  text, 
and  of  those  left  for  the  student  to 
work  out  for  himself,  is  extraordi- 
nary. From  this  point  of  view,  those 
teachers  and  pupils  who  are  accus- 
tomed to  or  prefer  a  different  text- 
book, would  still  do  well  to  provide 
themselves  with  this,  regarding  it 
merely  as  a  collection  of  examples 
and  without  any  reference  to  the  text. 


Elementary  Co-ordinate  Geometry. 

By  W.  B.  Smith,  Professor  of  Physics,  Missouri  State  University.  12mo. 
Cloth.    312  pages.    Mailing  Price,  $2.15;  for  introduction,  $2.00. 

VITHILE  in  the  study  of  Analytic  Geometry  either  gain  of 
*'  knowledge  or  culture  of  mind  may  be  sought,  the  latter 
object  alone  can  justify  placing  it  in  a  college  curriculum.  Yet  the 
subject  may  be  so  pursued  as  to  be  of  no  great  educational  value. 
Mere  calculation,  or  the  solution  of  problems  by  algebraic  processes, 
is  a  very  inferior  discipline  of  reason.  Even  geometry  is  not  the 
best  discipline.  In  all  thinking,  the  real  difficulty  lies  in  forming 
clear  notions  of  things.  In  doing  this  all  the  higher  faculties  are 
brought  into  play.  It  is  this  formation  of  concepts,  therefore,  that 
is  the  essential  part  of  mental  training.  And  it  is  in  line  with  this 
idea  that  the  present  treatise  has  been  composed.  Professors  of 
mathematics  speak  of  it  as  the  most  exhaustive  work  on  the  sub- 
ject yet  issued  in  America ;  and  in  colleges  where  an  easier  text- 
book is  required  for  the  regular  course,  this  will  be  found  of  great 
value  for  post-graduate  study. 


Wm.  G.  Peck,  Prof,  of  Mathe- 
matics and  Astronomy,  Columbia 
College :  I  have  read  Dr.  Smith's  Co- 
ordinate Geometry  from  beginning 
to  end  with  unflagging  interest.  Its 
well  compacted  pages  contain  an  im- 
mense amount  of  matter,  most  ad- 


mirably arranged.  It  is  an  excellent 
book,  and  the  author  is  entitled  to 
the  thanks  of  every  lover  of  mathe- 
matical science  for  this  valuable  con- 
tribution to  its  literature.  I  shall 
recommend  its  adoption  as  a  text* 
book  in  our  graduate  course. 


MATHEMATICS.  79 

Academic  Trigonometry :  piane  and  spherical. 

By  T.  M,  Blakslbb,  Ph.D.  (Yale),  Professor  of  Mathematics  in  the 
University  of  Des  Moines.  12mo.  Paper.  33  pages.  Mailing  Price, 
20  cents;  for  introduction,  15  cents. 

rpHE  Plane  and  Spherical  portions  are  arranged  on  opposite  pages. 

The  memory  is  aided  by  analogies,  and  it  is  believed  that  the 

entire  subject  can  be  mastered  in  less  time  than  is  usually  given  to 

Plane  Trigonometry  alone,  as  the  work  contains  but  29  pages  of  text- 

The  Plane  portion  is  compact,  and  complete  in  itself. 

Examples  of  Differential  Equations. 

By  George  A.  Osborne,  Professor  of  Mathematics  in  the  Massachu- 
setts Institute  of  Technology,  Boston.  12mo.  Cloth,  vii  +  50  pages. 
Mailing  Price,  GO  cents;  for  introduction,  50  cents. 

A  SP^RIES  of  nearly  three  hundred  examples  with  answers,  sys- 
■'^  tematically  arranged  and  grouped  under  the  different  cases, 
and  accompanied  by  concise  rules  for  the  solution  of  each  case. 

Selden  J.  Cofl&n,  lately  Prof,  of  I  Its  appearance  is  most  timely,  and  it 
Mathematics,    Lafayette     College  :  I  supplies  a  manifest  want. 

Determinants. 


The  Theory  of  Determinants:  an  Elementary  Treatise.  By  Paul  H. 
Hanus,  B.S.,  recently  Professor  of  Mathematics  in  the  University  of 
Colorado,  now  Principal  )f  West  High  School,  Denver,  Col.  8vo.  Cloth, 
viii  +  217  pages.    Mailing  Price,  $1.90;  for  introduction,  $1.80. 

rpHIS  book  is  written  especially  for  those  who  have  had  no  pre- 
vious knowledge  of  the  subject,  and  is  therefore  adapted  to 
self-instruction  as  well  as  to  the  needs  of  the  class-room.  The 
subject  is  at  first  presented  in  a  very  simple  manner.  As  the 
reader  advances,  less  and  less  attention  is  given  to  details. 
Throughout  the  entire  work  it  is  the  constant  aim  to  arouse 
and  enliven  the  reader's  interest,  by  first  showing  how  the  various 
concepts  have  arisen  naturally,  and  by  giving  such  applications  as 
can  be  presented  without  exceeding  the  limits  of  the  treatise.  The 
w^ork  is  sufficiently  comprehensive  to  enable  the  student  who  has 
mastered  the  volume  to  use  the  determinant  notation  with  ease, 
and  to  pursue  his  further  reading  in  the  modern  higher  algebra 
with  pleasure  and  profit. 


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